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Simplifying 3x(4x + 5) = 68 Reorder the terms: 3x(5 + 4x) = 68 (5 * 3x + 4x * 3x) = 68 (15x + 12x2) = 68 Solving 15x + 12x2 = 68 Solving for variable 'x'. Reorder the terms: -68 + 15x + 12x2 = 68 + -68 Combine like terms: 68 + -68 = 0 -68 + 15x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -5.666666667 + 1.25x + x2 = 0 Move the constant term to the right: Add '5.666666667' to each side of the equation. -5.666666667 + 1.25x + 5.666666667 + x2 = 0 + 5.666666667 Reorder the terms: -5.666666667 + 5.666666667 + 1.25x + x2 = 0 + 5.666666667 Combine like terms: -5.666666667 + 5.666666667 = 0.000000000 0.000000000 + 1.25x + x2 = 0 + 5.666666667 1.25x + x2 = 0 + 5.666666667 Combine like terms: 0 + 5.666666667 = 5.666666667 1.25x + x2 = 5.666666667 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = 5.666666667 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 5.666666667 + 0.390625 Combine like terms: 5.666666667 + 0.390625 = 6.057291667 0.390625 + 1.25x + x2 = 6.057291667 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 6.057291667 Calculate the square root of the right side: 2.461156571 Break this problem into two subproblems by setting (x + 0.625) equal to 2.461156571 and -2.461156571.Subproblem 1
x + 0.625 = 2.461156571 Simplifying x + 0.625 = 2.461156571 Reorder the terms: 0.625 + x = 2.461156571 Solving 0.625 + x = 2.461156571 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 2.461156571 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 2.461156571 + -0.625 x = 2.461156571 + -0.625 Combine like terms: 2.461156571 + -0.625 = 1.836156571 x = 1.836156571 Simplifying x = 1.836156571Subproblem 2
x + 0.625 = -2.461156571 Simplifying x + 0.625 = -2.461156571 Reorder the terms: 0.625 + x = -2.461156571 Solving 0.625 + x = -2.461156571 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -2.461156571 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -2.461156571 + -0.625 x = -2.461156571 + -0.625 Combine like terms: -2.461156571 + -0.625 = -3.086156571 x = -3.086156571 Simplifying x = -3.086156571Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.836156571, -3.086156571}
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